Geodesic Refinement Using James-Stein Estimators

In longitudinal imaging studies, geodesic regression in the space of diffeomorphisms [9] can be used to fit a generative model to images over time. The parameters of the model, primarily its initial direction or momentum, are important objects for study that contain biologically meaningful information about the dynamics occurring in the underlying anatomy. Unfortunately, it is common for any given subject to have a very limited number of longitudinal images available, the acquisition of which is corrupted by noise and variability due to scanning conditions. Furthermore, the underlying anatomy is subject to many entangled biological processes, the effect of which on images is in many cases poorly characterized. Hence, the approach must fit a model to few data points with uncharacterized variability. Here, we propose supplementing the lack of longitudinal information for an individual patient with information that can be extracted cross-sectionally from a population of time series to improve the model fit for the individual. To that effect, we propose a probabilistic model that leads to a well established technique from classical statistics: James-Stein estimators. We show that recent work on groupwise registration for improved geodesic estimation is a sub-optimal special case of our proposed model. Finally, we validate the model by showing geodesics refined by the James-Stein estimator extrapolate more accurately on average than raw geodesic estimates.